Comparative analysis. In Figure REF , we consider the case where \(\Sigma ^{(x)}\)  has a block structure and evaluate performance of the model under four scenarios: (A) including both \(\mbox{$\mu $}^{(t,m)}\)  and \(\Lambda \) , (B) ignoring only \(\mbox{$\mu $}^{(t,m)}\) , (C) ignoring only \(\Lambda \) , and (D) ignoring both \(\mbox{$\mu $}^{(t,m)}\)  and \(\Lambda \) . Ignoring \(\mbox{$\mu $}^{(t,m)}\)  corresponds to choosing \(\mbox{$\mu $}^{(t,m)} = \mbox{{$\mathbf {0}$}}\)  for all \(t\) ; ignoring \(\Lambda \)  implies choosing \(\Lambda = I_g\) . Scenario (A) refers to incorporating the structure as described in Section ; Scenario (B) is similar in spirit to other variable selection approaches [1]}, [2]} that incorporate dependence in the covariates through the indicator variable \(\zeta \)  (while ignoring the layer structure); Scenarios (C) and (D) correspond to vanilla Bayesian variable selection for each layer with uniform priors on selection probabilities for \(\zeta \)  [3]}. To establish comparison, we use the same data replications across all four scenarios. We see that incorporating the structure in \(X\)  through the hyperparameter \(\Lambda = cor(X)\) , and borrowing information from previous layers for the mean \(\mbox{$\mu $}^{(t,m)}\) , yields better estimation performance in terms of TPR, compared to having no structure by choosing \(\mbox{$\mu $}^{(t,m)} = \mbox{{$\mathbf {0}$}} ~\forall ~t\)  or \(\Lambda = I_g\) , or both. This improvement in performance holds true for all choices of the noise parameter \(\sigma ^2\) . Detailed results are shown in the second panel of Table S1. We have not included the false positive rates in the results as they were less than \(0.1\%\)  on average in all scenarios. We also compared the estimation under a multivariate-multiple regression with LASSO penalty, and observed high TPRs but also substantially higher false positive rates (for both Cases 1 and 2). Results presented in Table S1 report the averages obtained across 30 replications and the corresponding standard deviations. Details of the various hyperparameter choices in the data generation and model estimation processes are given in Section S3.
